 Solution

For a compact Lie group, take any positive-definite Hermitian inner product $h_0$ on $V$ and average using probability Haar measure:It remains positive definite, because the integrand is positive for a nonzero vector, and translation invariance makes it $G$-invariant. If $W\subset V$ is an invariant subspace, then $W^{\perp }$ is invariant too: $h(gv,w)=h(v,g^{-1}w)=0$ when $v\in W^{\perp }$. Thus $V=W\oplus W^{\perp }$. Induction on dimension proves complete reducibility. This uses the usual continuous finite-dimensional representations of the compact group.

For a connected complex reductive algebraic group, representations are understood to be rational representations. Use a compact real form $K\subset G$ with $\mathrm{Lie}(G)=\mathrm{Lie}(K){\otimes }_{\mathbb{R}}\mathbb{C}$. The structural construction takes a compact real form of the semisimple derived factor and the unit circles in the central torus, then passes through their finite central quotient. This is the compact-form structure of a complex reductive group, rather than an assumption that its given representation is already unitary.

The group $K$ is Zariski dense: the Lie algebra of its Zariski closure contains the complex span of $\mathrm{Lie}(K)$, hence all of $\mathrm{Lie}(G)$; connectedness then makes that closure $G$. Average over $K$ as above. The orthogonal complement of a $G$-submodule is $K$-invariant. Its stabilizer is a Zariski-closed subgroup, since preserving a fixed subspace is a system of algebraic matrix conditions. The stabilizer contains dense $K$, so it is all of $G$. This again gives an invariant complement to every submodule and provesIf reductivity is defined to allow finitely many components, first obtain a $G^0$-equivariant projection $P:V\to W$. Average its conjugates over the finite component group. Since $P$ commutes with $G^0$, the average is independent of coset representatives; it is $G$-equivariant and remains the identity on $W$. Its kernel supplies the required complement. Thus the result holds in that convention too.